budgeting for capital calls: a var-inspired approach

Budgeting forCapital Calls:A VaR-InspiredApproach

Luis O’Shea and Vishv Jeet

Burgiss Applied Research: Working Paper

March 2018

In this paper we discuss the importance, and challenges, offorecasting private capital cash flows. Particularly importantis forecasting capital calls, since they constitute liabilities forthe investor. While it is useful to know the expected capitalcalls arising from an investor’s portfolio, it is more importantto estimate a likely upper bound on those calls, since thiswill determine the reserves needed in order to safely servicethe calls. To this end, we introduce a new concept, namelythat of maximum probable contributions — a statistic which,subject to a user-specified confidence level, serves as such anupper bound. We explore in detail a historical methodologyfor its computation, illustrate typical model predictions, anddocument its out-of-sample performance during backtestingon both funds and portfolios of funds.

BUDGETING FOR CAPITAL CALLS 2

1 Introduction

Investors with commitments to private capital funds need to maintain sufficient reserves to service capitalcalls arising from those funds. If their reserves are insufficient, they may be unable to meet those capitalcalls (or perhaps, will be forced to sell illiquid assets at suboptimal prices). Conversely, excessive reserveswill reduce the amount of capital in the ground, and consequently will result in the investor forgoing some ofthe expected returns arising from private capital. This paper proposes a systematic way to strike a balancebetween the two extremes of insufficient reserves and excess caution.

Let us suppose an investor considers these questions with respect to a fixed horizon, such as one quarter:How much capital should one have on hand to service capital calls in the next quarter? A possible answerto this question is that one should keep on hand the expected 1 capital calls (henceforth, contributions) inthe next quarter. A moment’s thought, however, shows that this is insufficient. Consider a simple example:suppose it is equally likely that in the next quarter contributions will be either $2M or zero, then the expectedcontributions are $1M. Maintaining reserves of $1M means the probability that the investor will be unableto service their capital calls is 50%, a risk which is not likely to be acceptable. On the other hand, if onedemands that the risk of being unable to meet one’s capital calls be precisely zero, then the solution is simple:ensure reserves are at least equal to uncalled capital. This absolute certainty comes at a steep cost, namelythat of diluting the expected returns of one’s investments (since so much capital needs to kept in a liquidform). Informally it seems that a compromise is needed, whereby one keeps reserves at such a level thatcapital calls can usually be met, while allowing for the possibility that occasionally additional funds willbe needed.2 This idea, stated more formally, is the central concept discussed in this paper: given a level ofconfidence chosen by the investor (say, 95%), and a horizon (say, a quarter), we seek to estimate the amountof reserves necessary so that with probability 95% they will cover the capital calls in the next quarter. Wecall this amount the 95% maximum probable contribution (MPC) of the investor’s portfolio.

Our goal in this paper is to propose a methodology for computing MPC and to explore how the measureperforms. In section 2 we give an introduction to the topic of probabilistic bounds (such as MPC and VaR)and in section 3 we list the (essentially non-existent) literature related to this idea with regard to privatecapital. In section 4 we describe, in detail, our methodology for estimating MPC, in section 5 we illustratetypical model predictions for various fund types and ages, in section 6 we backtest our model, and in section 7we present our conclusions. Appendix B describes the data used for our model predictions and backtestingand appendix A provides more background on our backtesting methodology.

2 Risk and Probabilistic Bounds on Contributions

According to industry legend, around 1990 the chairman of JP Morgan requested that every day the risk ofthe bank’s trading portfolio be summarized in the form of a single number included in a report delivered tohis desk by 4:15pm. That number was the one-day 95% value-at-risk (VaR). The idea behind this number isthat while a portfolio’s value may be known today, at some future point in time (such as, tomorrow) its valuecan be best be described via a probability density function (PDF). Riskier portfolios would have more dispersePDFs than less risky ones. Thus the standard deviation of the PDF would indicate how risky the portfolio is.Indeed, if market returns were normally-distributed and all assets were linear, then the standard deviationwould suffice. However neither of these assumptions are correct (for example, options have payoffs which arenon-linear). These facts introduce the possibility of losses that exceed what one would anticipate based onstandard deviations alone. For example it is possible for two portfolios with equal standard deviations to havevery different probabilities of large losses. What JP Morgan’s chairman needed was a measure of tail risk.Thus the concept of VaR was born — a quantile from the predicted PDF of profits and losses. For example, on

1We are using the term expected in its statistical sense. For example, if one flips a coin (once) then the expected number ofheads is 0.5 (even though the observed number will always be either zero or one, and never one half!) One way of thinking ofthis is that it is the expected value is the long-run average. Equivalently, the expected value is the average over many identicalrealizations (coin flips, in the previous case). So the “expected capital calls” (over some period), are the average capital calls(over that period) from a large set of essentially identical funds.

2This compromise could be further rationalized as follows. Perhaps reserves are particularly liquid investments. If thoseliquid reserves prove insufficient then less liquid assets can be sold, perhaps at somewhat below market price. While evidentlyundesirable, this may be acceptable if it happens sufficiently rarely.

Burgiss Applied Research c© 2018 The Burgiss Group, LLC


19 out of 20 days (95% of the time) JP Morgan’s trading losses should be less than the reported one-day 95%VaR number.

Turning to private capital, investors face problems that are similar to those faced by JP Morgan (exceptrelated to cash flows rather than returns): the investor has a set of commitments which over the next quarter(say) will generate some set of capital calls. These calls cannot be known with certainty, thus the most one canhope for is a PDF of possible contributions (see figure 1 for an example of what such a PDF might look like).Note that such a distribution is very far from normal since contributions can only be positive, a contribution

0%

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Figure 1: PDF of contributions to a private capital portfolioThe above figure is a histogram of possible contributions to a private capital portfolio. The blue line representsthe expected contribution, the red line represents the 95% MPC contribution, and the red bars are contributionswhich exceed the MPC.

of precisely zero in any quarter is fairly common (i.e., the PDF is “zero-inflated”), and it decays to zero slowerthan a normal distribution.3 In the PDF of figure 1 one can see a large probability of contributions being zeroover the next period (this is the tall bar on the left of the histogram, and happens about 30% of the timein this example) as well as the exponential decay in probabilities for larger contributions. The expected (ormean) contribution is marked in blue. As can be seen, if one were to maintain reserves equal to that numberthen in about 50% of periods the reserves would be insufficient! Also shown (in red) is the 95th percentile(which is what we called the 95% MPC, above); by definition maintaining reserves at that level will result inthem being sufficient on 95% of all periods. However note that the 95% MPC is about twice the expectedcontribution; this illustrates the wide discrepancy between these two numbers. An investor could, of course,try to set the level of their reserves based on a combination of expected contributions and a rule of thumb; forexample, one could set reserves at twice the expected contributions. However, as we shall see in later sections,even this is likely to serve the investor poorly, on account of the fact that MPCs diversify across funds; thusdepending on the sizes and ages of the funds in a portfolio the multiple of expected contributions that oneshould employ will vary — perhaps doubling the expected cash flows will be appropriate for certain portfoliosat certain points of time, but in other situations it could be significantly different.

The above discussion should make clear that an estimate of MPC could be very useful for managing aportfolio of private capital funds. Furthermore computing such a measure is simple, assuming one has accessto the PDF of possible contributions in the next period (as illustrated in figure 1). Thus estimating MPC comesdown to predicting the PDF of future contributions. Much like when estimating VaR, there are two broadmethodologies for predicting such a PDF: historical and model-based. Briefly, the distinction between thesetwo approaches is as follows. In both VaR and MPC one must model how the market is likely to behave (VaR isconcerned with returns, MPC with cash flows). One can impose a parametric form on this behavior, in whichcase one must estimate the parameters of this form; this is a model-based methodology. Alternatively, one cansimply use historical data directly as a model for how the market behaves; this is a historical methodology. Inthis document we will focus almost exclusively on a historical methodology.

3Although it eventually goes to zero since contributions are capped by uncalled capital.



3 Literature Review

To the best of our knowledge MPC-like upper bounds to optimize capital reserves have not been mentioned orexplored in the published literature related to private capital cash flows. The closest is Meads et al. (2016)that highlights the importance of effectively managing capital reserves to avoid defaulting on capital calls.The authors consider historical quantiles of cumulative paid-in capital by fund age in the cross-section offunds but do not mention an MPC-like concept for future capital calls. Their analysis focuses on finding anoptimal division of the committed but uncalled capital between two different investment vehicles: Treasurysecurities and the public markets. For a detailed literature review on general cash-flow modeling we referreaders to our previous article (Jeet and O’Shea 2018) in which we explored advanced modeling choices forpredicting the expected size of future capital calls and showed a large improvement over the approach ofTakahashi and Alexander (2002).

4 Methodology

In this section we detail the methodology used to compute MPC. As mentioned in the previous section themethodology is what is often described as a historical methodology in the risk literature (Mina and Xiao 2001).The methodology has two components. First we describe how a PDF of cash flows is generated (sampling,below); this section is relatively novel. Second, we compute various statistics based on this PDF (statistics,below); this section is completely standard.

At a high level our approach is straightforward. Given a large database of historical fund cash flows, andgiven a fund we wish to analyze, we can find cash flows from funds which, at the time of the observation,were similar to our fund. This generates a pool of observations from which we can randomly sample a certainnumber of observations per period. Given a portfolio of funds we can carry out this procedure for each fund inthe portfolio and then aggregate the cash flows per sample. The result of this procedure is a PDF of portfoliocash flows drawn from the historical behavior of funds similar to those in our portfolio.

Next we describe the methodology more formally.

Sampling In order to define our methodology we first choose an analysis horizon (such as a quarter, or ayear). A period of time with length equal to the analysis horizon will be referred to as a period ; we will indexperiods by t. We also choose a particular such period, the analysis period, t0, to be the period for whichwe want to make predictions. For each period we also define the number of samples we will draw, nt (thisnumber will be relatively large for recent periods and will decline to zero at some point in the past). Next wechoose what we consider to be cash flows 4 in this methodology by defining either X = C (cash flows are justcontributions) or X = C −D (cash flows are contributions net of distributions).

We need a universe of funds for which we have attributes and cash flows; we will index these funds by i.Each fund has a set of, potentially time-dependent, attributes (such as subclass, size, age, uncalled capital);

we denote the attributes of the ith fund at the end of the tth period by A(i)t . Similarly we denote the cash

flows of the ith fund during the tth period by X(i)t .

Our methodology needs a notion of similarity between funds. This is defined in terms of fund attributes(for example perhaps we define two funds to be similar if they belong to the same subclass and if their agesdiffer by less than six months). In general, if a fund at time t (with attributes At) is considered similar toanother fund at time t′ (with attributes A′t′) then we write

At ∼ A′t′ .

Note that this notion of similarity, ∼, should be thought of as a (compound) parameter of the methodology,and in fact later we will discuss the trade-offs of different choices.

We start by defining the methodology for a single fund, the analysis fund, with attributes At. For eachperiod t we sample a number of cash flows from similar funds:

St,1, . . . , St,nt sampled randomly from{X

(i)t | A

(i)t ∼ At0 , i is any fund

}.

4Throughout this document we choose the sign of cash flows to be such that a positive cash flow indicates the flow of fundsfrom the investor into the fund. This is the opposite of the most common sign convention used by investors, but given our focuson capital calls, it is the most natural for this paper.



More generally we will have an analysis portfolio consisting of K funds. Corresponding to each of these fundswe will generate, as above, samples

S(k)t,1 , . . . , S

(k)t,nt

for k = 1, . . . ,K.

If these K funds have weight w1, . . . , wK in the portfolio, then we define our portfolio samples as

SPt,j = w1S(1)t,j + · · ·+ wKS

(K)t,j for j = 1, . . . , nt.

Finally we pool all these samples SPt,j (for all valid t and all valid j) to form a large set of cash flows which,thought of as a histogram, constitutes the historical prediction for the PDF of the cash flows of our analysisportfolio during the analysis period.

We call the parameters that are needed to define how sampling occurs — namely, the analysis horizon, theanalysis period, the number of samples per period, the definition of cash flows, and the similarity measure — thecash flow sampling settings (CFSS).

Statistics At this point we have a PDF of cash flows for our analysis portfolio:

S = {S1, . . . , Sn}

from which we can estimate various statistics. The MPC with confidence α (for example, α = 95%) is the αthquantile of S, namely the number s such that the fraction of elements of S that are less than s is α:

MPCα , smallest s such that the fraction for which S ≥ s is at least α.

We also computeexpected cash flows , meanS,

but note that this is the expected cash flows as estimated using the historical methodology. In general thisexpectation is better estimated using a model-based methodology and in fact is the topic of our previousworking paper (Jeet and O’Shea 2018).

Comments on the methodology Like all historical risk methodologies we need a large set of historicalobservations to ensure that the tails of the PDF are well-sampled (Pritsker 2001). As a result, the parameternt above, which dictates how many samples to draw from each historical quarter, and hence determines therelative weight of the recent and distant past, cannot decay too fast. In fact we typically keep nt constant fora couple of decades and then taper it down to zero quickly. Trying to let nt decay exponentially would riskreducing the sample size excessively (unless the half-life was made very long).

The requirements on large sample size also affect the similarity measure, ∼, defined above. As discussed,we sample from “similar” funds in the past, where we typically define two funds to be similar if they sharesubclass and have ages that differ by a relatively small amount. In principle one could also demand thatfunds be similar in terms of uncalled capital, or fund size. The danger, again, is that this could reducesample sizes excessively. Nevertheless, conditioning on some additional characteristics (especially uncalledcapital) seems appealing. One way to do this is to develop a parametric methodology,5 what we called amodel-based methodology above. In such a methodology the distributional assumptions would be explicitlyparametrized (for example, contributions could be exponentially distributed with a point mass at zero) andcertain parameters would need to be estimated. Such an approach would be less demanding in terms of dataand could make conditioning on further fund characteristics feasible. While we think that such a model-basedapproach is interesting, we will not discuss it further in this paper.

A further question one could ask is why, in the above sampling procedure, we maintain the identity ofquarters during portfolio formation, and only then pool samples to form the histogram that we use to computeour risk measures. Indeed, maintaining the identity of quarters is not necessary and dropping this requirementwould, in effect, increase our effective sample size (by allowing more mixing). However, there may be temporal

5We use the term parametric in the sense of parametric statistics. However note that an oft-cited document that describesVaR methodologies (Mina and Xiao 2001) uses the term parametric differently from us (they use it to mean that pricing functionsare linearized). To avoid confusion we refer to a parametric methodology as a model-based methodology.



correlations between cash flows, namely there may be periods when capital is called more rapidly and otherperiods when it is called more slowly, and furthermore this could be the same for the entire market. Bykeeping the identity of quarters we are preserving this correlation. Not doing so could potentially lead tounderestimating risk (i.e., underestimates of MPC).

5 Model Predictions

In this section we explore the typical behavior of the methodologies described in the previous section as wevary various fund and portfolio attributes.

Our methodology requires choosing values for a small set of parameters, the CFSS. We start by listing thevalues of those parameters used in this section:

Definition of cash flows Mostly we do not apply netting,6 so cash flows are defined to be contributions.However at times we will contrast predictions of contribution-only cash flows with predictions of netcash flows.

Analysis horizon We make predictions for a period of length one quarter.

Similarity measure Our historical methodology requires a similarity measure. In this section we definesimilar funds to be those that

• belong to the same subclass, and

• have ages that differ by less than three months.

Samples per period We use 30 samples per period for recent quarters (until 2003) then tapering down tozero (by 1993).

Analysis date We analyze our portfolio as of 2017 Q3.

Next we define our portfolio. It contains funds from five subclasses (buyout, venture capital, real estate,primary equity FoFs, and secondary equity FoFs). For each subclass the portfolio contains eleven funds of ages0, 1, 2, . . . , 10. Thus the portfolio contains a total of 55 funds. We commit one unit of capital to each fund, soour total commitment is 55. In general in the following discussion we will always predict the unscaled cashflows from this portfolio; in particular we will not be reporting cash flows as fraction of total commitment.

Cash flows PDFs Figure 2 contains various histograms that illustrate the result of applying our methodologyto our portfolio. We start our analysis by looking at the forecasted behavior of the entire portfolio (figure 2a).The colored vertical lines show the value of various measures computed for the entire portfolio: the prefix(“C.” or “N.”) indicate whether the statistics are computed using just contributions or contributions net ofdistributions. The suffixes “.90”, “.95”, “.99” denote various confidence levels for MPC and “.exp” denotesthe expected cash flow. Immediately one can see that MPCs are significantly greater than the expected cashflows; for example C.exp is a bit less than 1.5, while C.95 is about 2.25. The benefit of using net cash flows asopposed to just contribution grows as the portfolio contains more funds (this is because it is a diversificationeffect; distributions are very volatile, but in the presence of enough funds they become somewhat morepredictable and can offset more of the contributions). We can see that the N.95 is significantly lower than C.95.Finally, this plot also shows that the N.exp is negative (i.e., the portfolio is generating more distributionsthan absorbing contributions) which further illustrates how inappropriate expected cash flows are for settingreserve requirements. In contrast, all the net MPCs (N.90, N.95, and N.99) are positive, as expected.

Next, in figure 2b we break out cash flow PDFs for each subclass. Aside from the cash flows becomingsmaller (since each histogram corresponds to commitments to a single subclass), we can also see that thebenefit arising from using net cash flows, relative to just contributions, has declined. This can be seen bycomparing (say) N.95 (dashed green) with C.95 (solid green); in the top-level PDF the net MPC was abouthalf of the contribution-only MPC, while in these single-subclass portfolios the two are much closer.

6By netting we mean treating the cash flows that need to be forecast as the contributions net of the distributions in thatperiod. This is in contrast to ignoring the distributions and defining the cash flows to be just the contributions. See the start ofsection 4 for further details.



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(c) Fund-level cash flow PDFs for a sample of three fund ages (out of a total of 11)

Figure 2: Cash flow PDFs at various levels for a sample portfolio of 55 commitmentsThe horizontal axis represents unscaled cash flows, namely cash flows arising from a commitment of one unit toeach fund



Finally in figure 2c we plot PDFs for a subset of the funds. In this plot it is clear how the younger funds(say the 0Y funds) are calling more capital, but there is a significant uncertainty about just how much theywill call (thus in the figure the histograms appear to have an exponential tail to the right). In contrast theolder funds call much less and have a larger spike at zero (corresponding to the fact that in many quartersthey call precisely zero).


0.0 2.5 5.0 7.5 10.00.0 2.5 5.0 7.5 10.00.0 2.5 5.0 7.5 10.00.0 2.5 5.0 7.5 10.00.0 2.5 5.0 7.5 10.0

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Figure 3: One-quarter MPCs and expected cash flows of a single fund, as a function of fund ageIn the above figure the 90%, 95%, and 99% MPCs are plotted in green, blue, and red. The expected cash flows areplotted in black. (The shaded ribbon indicates a 95% confidence interval on the quantile estimators, but shouldbe interpreted as an approximate lower bound on errors.)

MPC as a function of age Figure 3 shows how the MPC of a single fund varies as the fund ages. In thatfigure we can see various stylized facts about fund behavior. First, note that for all subclasses very youngfunds have a very sharp MPC peak at inception (zero age); this is partially the nature of how these fundscall capital, but is also partially the result of the Burgiss Manager Universe (BMU) defining the inception ofa fund to be when its first cash flows occurs. Real estate has very large MPCs early on (especially for highconfidence levels) since many such funds call most of the capital very early on. In contrast primary equityFoFs have gentler spikes early on, as expected since the FoF itself must first commit capital before any capitalcalls can be passed through to the downstream LPs.

Contributions versus net contributions Figure 4 compares contribution-only with net contribution 95%MPCs for a single fund from each subclass. Not surprisingly the net MPC is smaller than contribution-onlyMPC, however for regular funds (not FoFs) the difference is marginal. For FoFs using net cash flows makes amore significant difference. The reason for this is that distributions are much less predictable (in the sense ofbeing much more disperse). This means that while the expected net cash flows may (for old enough funds)become negative (i.e., become a net distribution) the MPC is unable to leverage those distributions since theyare too disperse. Since FoFs are, in effect, a portfolio of funds their distributions are diversified and hence lessdisperse; consequently they offset the contributions in a more predictable way and hence bring down the MPC

to a greater degree.




0.0 2.5 5.0 7.5 10.00.0 2.5 5.0 7.5 10.00.0 2.5 5.0 7.5 10.00.0 2.5 5.0 7.5 10.00.0 2.5 5.0 7.5 10.0

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Figure 4: One-quarter 95% MPCs and expected cash flows of a single fund, as a function of fund ageIn the above figure the 95% MPCs are plotted in blue. The expected cash flows are plotted in black. The dashedlines are the contribution-only MPCs, while the solid lines are net cash flow MPCs.


1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

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Figure 5: One-quarter cash flow statistics for an increasingly diversified portfolioThe plot shows the expected cash flows and MPCs (at various confidence levels) of a portfolio of distinct funds,each of age 1 year, and drawn from the same subclass (indicated by the facet label). The total commitment iskept at 1 but it is distributed over an increasing number of funds.



MPC as a function of number of funds Figure 5 plots how the MPC of a portfolio of funds varies as thenumber of funds in the portfolio is increased. In this figure we always have a fixed commitment (of 1.0)however we assume that we have divided that commitment equally among n funds of age 1Y (as of analysis).These funds, although identical (in terms of age and subclass), are distinct and hence diversify each other.Start by noting that the expected contributions (black line) are constant. This is because the expectation ofthe portfolio is the sum of the expectations of its constituents.7 Again this serves to illustrate unsuitability ofcash flow expectations for the purposes of setting capital reserves. What one would expect is that a singlefund would be much less predictable than a portfolio of such funds, and that one would need more reserves.Indeed, plotting MPCs, this is exactly what we see. Furthermore, we see a clear difference between FoFs andfunds. The decline in the MPCs of portfolios of FoFs tapers off at about 2–3 FoFs for secondary FoFs and atabout 3–4 FoFs for primary FoFs. In contrast portfolios of funds continue to exhibit diversification (decliningMPCs) for larger numbers of constituents in the portfolio. This is exactly what one would expect given thatFoFs are already a portfolio of funds.

An interesting feature of figure 5 is that the MPC (for any confidence level) does not appear to be convergingto the expected cash flows (the black line). It might at first seem that this should occur (as a consequenceof the central limit theorem). The observed behavior is explained by two effects. First, the convergence isrelatively slow (the gap should be approximately inversely proportional to the square root of the number offunds). Second, there is a common factor driving all contributions. As a result of this factor contributions arepartially cross-sectionally correlated. In fact, this is precisely why in our methodology we chose to maintainquarter identity during portfolio formation, and only pool at the very end. We feel this is an important featureof this methodology.

6 Backtesting

It is important to understand how accurate our estimates of MPC are in practice. For this purpose weperformed a backtesting analysis for individual funds as well as portfolios of funds. The backtesting procedurewe employ is standard.8 For a given fund and a quarter we predict a PDF for the next quarter’s contributionsusing the methods in section 4. We use the same CFSS listed at the start of the section 5, except we used asimilarity measure with an age window of six month (instead of three) and we only allowed data two quartersbefore the analysis quarter into the PDF prediction. For example if we are predicting MPCs for 2017 Q3 wedo not use data from both Q3 and Q2 of 2017. The reason for this is to mirror the typical data-productiondelays due to lagged reporting in practice.

6.1 Fund-Level Backtests

Figure 6 displays a representative fund-level backtest that ran for 40 quarters and compare the performanceof several MPCs with the observed contribution following their prediction. One can see that whenevera contribution was made, its magnitude almost always exceeded the expected contribution (displayed inpink). The median estimate (displayed in red) did even worse because it was almost always zero due tozero-inflation (ZIF) in contributions. Clearly neither the median nor the expected contributions serve as areasonable estimate of needed capital reserves. For this particular backtest the 90% MPC would have donemuch better. In principle a 90% MPC will be exceeded, in long run, only 10% of time; similarly a 95% MPC

will be exceeded only 5% of time. Note too that for this backtest the highest value of 95% MPC would havebeen just 30% of the commitment (near inception).

For our fund-level backtesting analysis we ran more than three thousand backtests; one per fund forfunds in three major subclasses in the US private capital market: buyout, real estate, and venture capital.These backtests generated a predicted PDF for each observed contribution across all funds and quarters. Wesummarize this data in two different ways so that it is easy to comprehend, visualize, and gain insights.

One simple way to summarize an observed contribution and its predicted PDF is to compute a binary-valuedmeasure called excession. An excession is assigned a one if the observed contribution exceeds an MPC (say95%), and a zero otherwise. This way the theoretical PDF of excessions would be binomial. For instance,

7More formally, the expectation operator is linear.8For more details on our backtesting set up we refer readers to appendix A.



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Figure 6: A sample fund-level backtestThe blue bars represent the actual quarterly contributions and solid lines are smoothed time-series MPCs. Notethat the negative bar is, in fact, a contribution but represents returned capital.

say in a given quarter there are 100 funds, for 95% MPC we would expect Binomial(100, 0.05) excessions.9

Similarly in the temporal direction, say contributions of a single fund over 40 quarters, for 95% MPC we wouldexpect Binomial(40, 0.05) excessions.

Another more intelligent way to summarize this data is to compute a quantile rank of the observedcontribution from the predicted PDF. For sufficiently large number of observations we can expect thesequantiles to be roughly uniformly distributed between zero and one.10

Figures 7 and 8 plot the PDF of excessions observed both in temporal and cross-sectional dimensions.The theoretical PDF is also included for comparison (in red). One can note that the two PDFs match verywell in the cross-section of funds but not so much in the temporal dimension. This is expected and can beexplain by two factors. First, the temporal dimension has fewer observations (90 quarters) as opposed to thecross-sectional dimension that has several hundred observations (funds) in each subclass. The second, anda deeper reason, is the business-cycle variability across time that is inherent in the observed data but it ismissing from the theoretical model.

Figure 9 display the data of figure 7 in a time-series fashion. This is useful because this plot can detect theeffectiveness MPCs over the long-run business cycles and during broad-market events. We observe that mostdata points stay close to their expected values (shown in black horizontal lines at 10%, 5%, and 1%) andwithin the two standard deviation interval (gray colored area) but there are some large deviations (venturecapital in particular) around times of market crises. There appears to be an absence of excessions during thebroad-market crises. It is possible that most of the time variations in the frequency of excession (blue dotsfalling outside the gray area) are due to continuously changing market conditions before, during, and afterthe two major broad-market events: dot-com crash (DTC) and global financial crisis (GFC).

Figure 10 displays the histogram of quantile ranks produced in our fund-level backtesting analysis. Giventhat the quarterly fund-level contribution data is very noisy, the histogram of these quantile ranks is remarkablyclose to uniformly distributed. The low density for the lower quantiles, and therefore high density for thehigher quantiles, is perhaps due to the some interplay of three issues: the noise in the contributions data, the

9A note on notation: Binomial(N, p) is an integer-valued random number drawn from a binomial distribution with meanN × p.

10Appendix A provides a good discussion on comparing excession and quantile rank.



Buyout Real Estate Venture Capital90%

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Figure 7: Comparison of theoretical and observed binomial PDF across quartersThe red line represents theoretical and the blue line (and bars) represents the observed PDF.

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Figure 8: Comparison of theoretical and observed binomial PDF across fundsThe red line represents theoretical and the blue line (and bars) represents the observed PDF.



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Figure 9: Average excession across timeThis plot is useful to detect clustering of (or absence of) excession events across time.

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Figure 10: Histogram of quantile ranks of quarterly fund-level quarterly contributionsThe theoretical expectation of this histogram is the uniform PDF represented by a dashed line. The shaded areain blue covers the 95% confidence interval for the height of each quantile bar. The three vertical lines in red,green, and blue represent 90%, 95%, and 99% quantiles respectively.



ZIF problem, and the limitations of the historical-simulation methodology. The expectation that quantileranks be distributed uniformly does not take into account ZIF. The presence of ZIF distorts the PDF of quantileranks and requires ZIF-sensitive quantile-ranking methods, see appendix A.1 for more details.

6.2 Portfolio Backtests

Next we move to backtesting of portfolios of private capital funds. For this analysis we build random portfolioswith a simple policy of investing in one fund per vintage from 1995 to 2017. In each vintage the selected fundcould come from either buyout, venture capital, or real estate subclass. All selected funds are weighted equallyto form a portfolio. Our analysis is based on 1000 such portfolios sampled randomly. Figure 11 displaysbacktest of one such portfolio. Note that a portfolio of funds almost always makes contributions every quarter.Similar to what we saw in fund-level backtest that the expected contribution limit is frequently exceeded bythe actual amount of contributions following prediction. For the particular portfolio shown in figure 11, the95% MPC was exceeded only three times over a period of 22 years. What is more interesting is that portfoliosenjoy significant diversification benefit as well, which means that capital reserves are shared among the fundsthat are actively calling capital. A portfolio that invests in one fund per vintage may have, in the long run, 5to 7 active funds11 that can potentially call capital, yet the capital reserves requirement set by 95% MPC isless than the commitment into a single fund.

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Figure 11: A sample portfolio-level backtestThe blue bars represent the actual quarterly contributions and solid lines are smoothed time-series MPCs.

The observations we made above are specific to the portfolio selected for figure 11. In order to test theirstatistical significance we plot summary information from these backtests in figures 12 and 13. Figure 12compares expected contributions with three MPCs levels: 90%, 95%, and 99%. We plot the density of excessionmagnitude and color the portion (in red) where the excession is positive, i.e., the percent of times an observedcontribution exceeded the stated MPC level. One can see that across all contribution events, observed for1000 portfolios over 90 quarters, expected contribution was exceeded at the rate of almost 50% but the MPCs

correctly exceeded only at the rate that is pre-defined, i.e., 5% for 95% MPC and so on.Figure 13 plots the histogram of the quantile ranks produced in our portfolios backtesting analysis and as is

expected the histogram roughly matches the uniform PDF. Unlike the fund-level contributions data (shown in

11Since most funds call most of the committed capital within first 5 years.



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Figure 12: Density plots of excession magnitudeThe area on the right-hand side of the dashed-line roughly represent the probability of exceeding the underlyingMPC.

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Figure 13: Histogram of quantile ranks of quarterly portfolio-level quarterly contributionsThe shaded area in blue covers the 95% confidence interval for the height of each quantile bar. The three verticallines in red, green, and blue represent 90%, 95%, and 99% quantiles respectively.



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Figure 15: Average excession across timeThis plot is useful to detect clustering of (or absence of) excession events across time.



figure 10) the portfolio-level contributions data are relatively less noisy and do not have the ZIF problem. Thisexplain why the histogram in Figure 13 is much closer to the theoretical uniform distribution (represented bydashed black line) than those in figure 10. Although looking at the higher MPCs (95% onward) in Figure 13,it seems they were overestimated. We think this is perhaps due to the limitations of historical-simulationsmethodology and reporting delays.12

Figure 14 plots the PDF of excession observed both in temporal and cross-sectional dimensions for portfoliocash flows, the theoretical PDF is also included for comparison (dashed line in red), similar to what we saw inthe fund-level backtesting analysis. The empirical and theoretical PDFs are generally in agreement but withsome mismatch in the temporal dimension perhaps, again, because it has fewer observations and the theoreticalmodel does not capture the business-cycle variability inherent in the observed data. Finally figure 15 plotsthe mean excession in a time-series fashion to detect any clustering (or absence) of excession events duringspecific period. Once again, there appears to be an absence of excessions during the broad-market crises,specially the GFC.

7 Conclusion

This paper is concerned with making risk predictions about future contributions. A risk prediction is onewhere what is predicted is not the value of some random variable (such as the total capital calls arising from aportfolio) but the distributional properties (i.e., the PDF) of that variable. Once such a PDF is predicted it isstraightforward to calculate various measures, such as quantiles, that serve as probabilistic upper bounds onthe contributions in the next period. For example, the 95% MPC is an amount such that the contributions inthe next period will be less than that amount with probability 95%. These quantiles are actionable numbersfor investors. For example, investors that maintain reserves at the 95% quarterly MPC can expect to onlyneed funds in excess of those reserves once every five years; maintaining reserves at 99% MPC should resultin this happening once every 25 years. In contrast, actual contributions will exceed expected contributionsroughly twice a year.

We propose a methodology based on historical sampling for estimating the PDF of these contributions,which in turn allows us to estimate MPCs. This methodology takes into account, in a natural and quantitativeway, many expected effects including: the age-dependent rate at which funds call capital, the variation acrosssubclass in how funds call capital, the diversification benefit arising from portfolios that have multiple funds,and the reduction in MPC if contributions are offset against distributions in each period. After illustratingthese effects, we also backtested the performance of the risk model and found it to perform remarkably well.The backtesting was carried out against both funds and random portfolios and by simultaneously looking atmultiple confidence levels.

In summary, historically-estimated MPCs seem to be accurate, quantitative, and actionable measures ofmuch contributions an investor can expect in the next period. Furthermore MPCs capture many complexinteractions between the cash flows of private capital funds, including their age dependence and the degreethat they diversify among each other.

12In general backtesting only uses data strictly before the observation it is trying to predict. However we skip an additionalquarter of data since we think that on account of reporting delays this better reflects the historical data that would be availableto any model making predictions.



References

Jeet, V. and L. O’Shea (Jan. 2018). Modeling Cash Flows for Private Capital Funds. Working Paper 4. TheBurgiss Group LLC.

Meads, C., N. Morandi, and A. Carnelli (2016). “Cash Management Strategies for Private Equity Investors”.In: Alternative Investment Analyst Review 4 (4), pp. 31–43.

Mina, J. and J. Xiao (2001). Return to RiskMetrics: The Evolution of a Standard. Tech. rep. RiskMetricsGroup.

Pritsker, M. G. (2001). The Hidden Dangers of Historical Simulation. Working Paper. The Federal ReserveBoard.

Takahashi, D. and S. Alexander (2002). “Illiquid Alternative Asset Fund Modeling”. In: The Journal ofPortfolio Management 28.2, pp. 90–100.

Zumbach, G. (2007). Back testing risk methodologies from one day to one year. Tech. rep. RiskMetrics Group.



A Backtesting Risk Methodologies

In this document a risk prediction is a prediction of the distribution (PDF) of a random variable (rather thanjust its value). A risk methodology is a technique for generating these forecasted PDFs. In general these fallinto two classes: model-based (where the PDF belongs to some family and the model’s job is merely to estimatethe parameters for an instance from that family) and historical (where the PDF is derived non-parametricallyfrom historical data).

A common approach to backtesting a risk methodology is to derive a tail measure, such as the 95thpercentile, and then count excessions, namely observations that exceed the tail measure. A good model, inthis case, will have an excession rate of about 5%. Note the excession rates that are too high or too lowindicate deficiencies in the risk model. A disadvantage of this approach is that it focuses on a single confidencelevel. If one were interested in a different level (such as 99% MPC) then the entire backtest would need tobe repeated. In addition this approach discards information from “near misses” since one can only countexcessions, there is no such thing as a “small” excession or an “almost” excession.

These problems can be avoided by instead focusing on a different variable, namely transforming eachobservation into its quantile rank via the predicted PDF (Zumbach 2007). For example, suppose that ateach period t we observe a real-valued random variable Xt. Corresponding to each observation we make aprediction for the PDF of X at t, φt(·). Let Φt(·) be the corresponding cumulative density function (CDF),then qtΦt(Xt) is a number between 0 and 1. Furthermore, if each Xt is, in fact, distributed according to φt(so the predictions were perfect) then qt is a uniform random variable between 0 and 1. Furthermore if φt isthe true distribution of Xt conditional on all information up to t then qt should be temporally uncorrelated.

For example, suppose Xt is a standard normal random variable, and suppose we consider three models (orpredictions) for its PDF: normal distributions with mean zero but with standard deviations of 1 (correct), 0.8(under-predicting risk), and 1.2 (over-predicting risk). The distributions of qt for each of these models areillustrated in figure 16. The top facet (labeled “correct”) shows the result of using the correct model. As

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Figure 16: Probability of getting various quantile ranks for several illustrative modelsIn this figure the data and models are normally distributed with mean zero. The data has a standard deviation of1 and the models have standard deviations of 1, 1.2, and 0.8. The number of data points is 104. The horizontalribbon is a 95% confidence interval around the expected density of quantile ranks, assuming the model is correct.

expected the quantile ranks are approximately uniformly distributed between 0 and 1. Note too that there isa certain amount of natural variation (or noise) arising from the fact that the histogram is based on a finite



sample of data. The horizontal ribbon in the figure is a 95% confidence interval on the number of quantileranks in each bar of the histogram. As can be seen, the correct model stays within the confidence intervalabout 95% of the time. In contrast the model that over-predicts risk has a smaller number of quantile ranksthan expected at both end of the graph (and strays far outside the confidence interval). Similarly, the modelthat under-predicts risk has too many quantile ranks at each end of the graph and again strays far outsidethe confidence interval.

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Figure 17: CDF and quantile ranks arising from a distribution with a point massThe distribution is a normal distribution with mean 1, but with an additional point mass at zero.

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A.1 Distributions with Point Masses

The discussion in the previous section regarding quantile ranks assumed that the model distribution iscontinuous. If the model distribution has point masses13, 14 then the above procedure runs into difficulties.Consider the CDF depicted in figure 17a; it corresponds to a normal distribution with mean 1 and with apoint mass at zero. What should the quantile rank corresponding to x = 0 be? Choosing either the lower(red) or upper (blue) quantile rank will result in distributions of quantile ranks that are far from uniform. Forexample, in figure 18 we have plotted the distribution of quantile ranks resulting from these two choices (inthe facets labeled ‘upper’ and ‘lower’) which clearly show a non-uniform distribution, even though our modelis correct! In our backtesting we employ a simple fix for this issue. Instead of choosing the upper or lowerquantile we choose a quantile uniformly at random between these two extremes. This choice is displayed infigure 17b where the red points have been distributed randomly between the possible extremes. The facetlabeled ‘random’ in figure 18 shows the result of this fix, which as can be seen makes the quantile ranksuniformly distributed as expected when the model agrees with the data.

B Data used for Model Estimation and Backtesting

All computational results in this paper are based on private capital data in the BMU as of 2017 Q3.15 Our dataconsisted of US funds denominated in USD (excluding funds of funds) from three subclasses of funds, namelybuyout, real estate, and venture capital. For FoFs analysis we focus on US FoFs denominated in USD that arefurther categorized as primary or secondary in their market focus. Both funds and FoFs are distributed over abroad range of vintage years from 1980 to 2017. We aggregated dated Contributions and Distributions to theend-of-quarter date for each fund or FoF.

13For a purely continuous distribution the probability of observing any particular value (as opposed to an interval) is zero.However if the probability of observing some x is non-zero, then the distribution is said to have a point mass at x.

14Note that the PDF of private capital cash flows does have a point mass (at zero) since there will be many quarters with nocash flows.

15The Burgiss Manager Universe is a research-quality dataset comprised of nearly 40 years of daily cash flows and valuationsfor over 7,800 private capital funds, representing more than $5 trillion of capital committed across the globe in various PrivateEquity, Private Debt and Real Asset strategies. The dataset covers a full spectrum of strategies across all geographies, and BMU

data is representative of global institutional investor experience because it is sourced entirely from limited partners, avoiding thenatural biases associated with other data sourcing models.



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budgeting for capital calls: a var-inspired approach

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